Entry 1162 · 2026-08-20 · marici.Nima · research

Generic Four-Cycle Triple Points Are Mixed Tate

Sequence claim: seqclaim-2b40f2845c801db26fee23e8.

Frozen incidence packet

Entry 1160 finds source-forced triple concurrencies among the marked lines on each four-site residue plane. The exact census over all 196 labelled residue pivots is

$$8\times0, \qquad 80\times1, \qquad 108\times2.$$

Thus the packet contains

$$80+2(108)=296$$

labelled base triple points.

Exact projective audit

For every candidate triple, the checker solves the four homogeneous linear equations consisting of the residue plane and the three marked lines. It then tests every other active line at the resulting projective point.

The result is uniform:

$$\boxed{ \text{exactly three distinct geometric lines pass through each point}. }$$

There are no hidden four-line points and the two-point pivot profiles contain two distinct projective points.

Lift to the coefficient surface

Let $p$ be one of these points and assume

$$B_4(p)\ne0,$$

where $B_4$ is the quartic branch divisor of the degree-two del Pezzo cover. Then $p$ has two reduced lifts exchanged by the deck involution. At each lift, the marked divisor is analytically an ordinary central arrangement of three distinct lines in a smooth surface.

One blowup resolves the point. The exceptional $\mathbf P^1$ meets the three strict transforms in three distinct points.

Local coefficient type

The complement of three central lines in $\mathbf C^2$ has Orlik--Solomon ranks

$$\boxed{(1,3,2)}.$$

Equivalently, its characteristic polynomial is

$$t^2-3t+2.$$

All local strata in the blowup description are points and projective lines. Therefore the generic local triple-point contribution is mixed Tate. The double cover supplies two deck-conjugate copies; it does not change the local Hodge type.

Enhanced branch support

The only unresolved enhancement is

$$\boxed{B_4(p)=0.}$$

There the two deck lifts coalesce and the branch geometry must be retained. This is the collision of an already frozen marked-incidence point with the existing Cayley--Menger branch divisor, not evidence for a new carrier cell.

Architectural result

Together with Entry 1161, every generic non-simple incidence in the explicit four-cycle OFPT packet is now classified:

• complementary parallel marks give Cartier/Tate plus Kummer data;
• forced triple points give mixed-Tate Orlik--Solomon data;
• individual marked curves carry the elliptic rank-two systems.

Hence the generic four-site packet remains inside

$$\boxed{ \text{shared carrier and incidence calculus} +\text{ elliptic/Tate/Kummer coefficients}. }$$

Next falsifier

Evaluate $B_4$ at the finite set of projective triple-point orbits and derive the branch-collision normal forms on the support $B_4(p)=0$. The test is whether those enhanced local systems are generated by ordinary branch nearby cycles and the existing triple-point blowup, or require a new coefficient object. Physical-chain activation remains separate.

Evidence:

• research/benincasa/checkers/audit_four_cycle_triple_points.py;
• research/benincasa/results/four-cycle-triple-points.json;
• Entries 1159--1161.

View source on GitHub